Showing papers on "Potts model published in 2009"

A convex relaxation approach for computing minimal partitions

Abstract: In this work we propose a convex relaxation approach for computing minimal partitions. Our approach is based on rewriting the minimal partition problem (also known as Potts model) in terms of a primal dual Total Variation functional. We show that the Potts prior can be incorporated by means of convex constraints on the dual variables. For minimization we propose an efficient primal dual projected gradient algorithm which also allows a fast implementation on parallel hardware. Although our approach does not guarantee to find global minimizers of the Potts model we can give a tight bound on the energy between the computed solution and the true minimizer. Furthermore we show that our relaxation approach dominates recently proposed relaxations. As a consequence, our approach allows to compute solutions closer to the true minimizer. For many practical problems we even find the global minimizer. We demonstrate the excellent performance of our approach on several multi-label image segmentation and stereo problems.

Multiresolution community detection for megascale networks by information-based replica correlations.

Abstract: We use a Potts model community detection algorithm to accurately and quantitatively evaluate the hierarchical or multiresolution structure of a graph. Our multiresolution algorithm calculates correlations among multiple copies ("replicas") of the same graph over a range of resolutions. Significant multiresolution structures are identified by strongly correlated replicas. The average normalized mutual information, the variation in information, and other measures, in principle, give a quantitative estimate of the "best" resolutions and indicate the relative strength of the structures in the graph. Because the method is based on information comparisons, it can, in principle, be used with any community detection model that can examine multiple resolutions. Our approach may be extended to other optimization problems. As a local measure, our Potts model avoids the "resolution limit" that affects other popular models. With this model, our community detection algorithm has an accuracy that ranks among the best of currently available methods. Using it, we can examine graphs over 40 x10;{6} nodes and more than 1 x10;{9} edges. We further report that the multiresolution variant of our algorithm can solve systems of at least 200 000 nodes and 10 x 10;{6} edges on a single processor with exceptionally high accuracy. For typical cases, we find a superlinear scaling O(L1.3) for community detection and O(L1.3 log N) for the multiresolution algorithm, where L is the number of edges and N is the number of nodes in the system.

Algorithms for entanglement renormalization

Abstract: We describe an iterative method to optimize the multiscale entanglement renormalization ansatz for the low-energy subspace of local Hamiltonians on a $D$-dimensional lattice. For translation-invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of one-dimensional systems, namely, Ising, Potts, $XX$, and Heisenberg models. The potential to compute expected values of local observables, energy gaps, and correlators is investigated.

Conditions for Rapid Mixing of Parallel and Simulated Tempering on Multimodal Distributions

Abstract: We obtain upper bounds on the convergence rates of Markov chains constructed by parallel and simulated tempering. These bounds are used to provide a set of sucien t conditions for torpid mixing of both techniques. We apply these conditions to show torpid mixing of parallel and simulated tempering for three examples: a normal mixture model with unequal covariances in R M and the mean-eld Potts model with q 3, regardless of the number and choice of temperatures, and the meaneld Ising model when an insucien t set of temperatures is chosen. The latter result contrasts with the rapid mixing of parallel and simulated tempering on the meaneld Ising model with a linearly increasing set of temperatures as shown previously.

Simulating quantum computers with probabilistic methods

Abstract: We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove that sparse quantum circuits as well as circuits composed of CNOT and exp[iaX] gates can be simulated classically. In a second part, we apply our results to the simulation of quantum algorithms. It is shown that a recent quantum algorithm, concerned with the estimation of Potts model partition functions, can be simulated efficiently classically. Finally, we show that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes. Specifically, we prove that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.

Ordered states of adatoms on graphene

Abstract: We show that a dilute ensemble of epoxy-bonded adatoms on graphene has a tendency to form a spatially correlated state accompanied by a gap in graphene's electron spectrum. This effect emerges from the electron-mediated interaction between adatoms with a peculiar $1/{r}^{3}$ distance dependence. The partial ordering transition is described by a random bond three-state Potts model.

Variational Bayes for estimating the parameters of a hidden Potts model

Abstract: Hidden Markov random field models provide an appealing representation of images and other spatial problems. The drawback is that inference is not straightforward for these models as the normalisation constant for the likelihood is generally intractable except for very small observation sets. Variational methods are an emerging tool for Bayesian inference and they have already been successfully applied in other contexts. Focusing on the particular case of a hidden Potts model with Gaussian noise, we show how variational Bayesian methods can be applied to hidden Markov random field inference. To tackle the obstacle of the intractable normalising constant for the likelihood, we explore alternative estimation approaches for incorporation into the variational Bayes algorithm. We consider a pseudo-likelihood approach as well as the more recent reduced dependence approximation of the normalisation constant. To illustrate the effectiveness of these approaches we present empirical results from the analysis of simulated datasets. We also analyse a real dataset and compare results with those of previous analyses as well as those obtained from the recently developed auxiliary variable MCMC method and the recursive MCMC method. Our results show that the variational Bayesian analyses can be carried out much faster than the MCMC analyses and produce good estimates of model parameters. We also found that the reduced dependence approximation of the normalisation constant outperformed the pseudo-likelihood approximation in our analysis of real and synthetic datasets.

Reconstruction for the Potts model

Abstract: The reconstruction problem on the tree plays a key role in several important computational problems. Deep conjectures in statistical physics link the reconstruction problem to properties of random constraint satisfaction problems including random k-SAT and random colourings of random graphs. At this precise threshold the space of solutions is conjectured to undergo a phase transition from a single collected mass to exponentially many small clusters at which point local search algorithm must fail. In computational biology the reconstruction problem is central in phylogenetics. It has been shown [Mossel 04] that solvability of the reconstruction problem is equivalent to phylogenetic reconstruction with short sequences for the binary symmetric model.Rigorous reconstruction thresholds, however, have only been established in a small number of models. We confirm conjectures made by Mezard and Montanari for the Potts models proving the first exact reconstruction threshold in a non-binary model establishing the so-called Kesten-Stigum bound for the 3-state Potts model on regular trees of large degree. We further establish that the Kesten-Stigum bound is not tight for the $q$-state Potts model when q ≥ 5. Moreover, we determine asymptotics for these reconstruction thresholds.

On the Existence of Generalized Gibbs Measures for the One-Dimensional p-adic Countable State Potts Model

Abstract: We consider the one-dimensional countable state p-adic Potts model. A construction of generalized p-adic Gibbs measures depending on weights λ is given, and an investigation of such measures is reduced to the examination of a p-adic dynamical system. This dynamical system has a form of series of rational functions. Studying such a dynamical system, under some condition concerning weights, we prove the existence of generalized p-adic Gibbs measures. Note that the condition found does not depend on the values of the prime p, and therefore an analogous fact is not true when the number of states is finite. It is also shown that under the condition there may occur a phase transition.

Modulated Phase of a Potts Model with Competing Binary Interactions on a Cayley Tree

Abstract: We study the phase diagram for Potts model on a Cayley tree with competing nearest-neighbor interactions J1, prolonged next-nearest-neighbor interactions Jp and one-level next-nearest-neighbor interactions Jo. Vannimenus proved that the phase diagram of Ising model with Jo=0 contains a modulated phase, as found for similar models on periodic lattices, but the multicritical Lifshitz point is at zero temperature. Later Mariz et al. generalized this result for Ising model with Jo≠0 and recently Ganikhodjaev et al. proved similar result for the three-state Potts model with Jo=0. We consider Potts model with Jo≠0 and show that for some values of Jo the multicritical Lifshitz point be at non-zero temperature. We also prove that as soon as the same-level interactionJo is nonzero, the paramagnetic phase found at high temperatures for Jo=0 disappears, while Ising model does not obtain such property. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work by Ganikhodjaev et al. for Jo=0. At vanishing temperature, the phase diagram is fully determined for all values and signs of J1,Jp and Jo. At finite temperatures several interesting features are exhibited for typical values of Jo/J1.

Restricted orientation "liquid crystal" in two dimensions: Isotropic-nematic transition or liquid-gas (?)

Abstract: We present Monte Carlo simulation results of the two-dimensional Zwanzig fluid, which consists of hard line segments which may orient either horizontally or vertically. At a certain critical fugacity, we observe a phase transition with a two-dimensional Ising critical point. Above the transition point, the system is in an ordered state, with the majority of particles being either horizontally or vertically aligned. In contrast to previous work, we identify the transition as being of the liquid-gas type, as opposed to isotropic-to-nematic. This interpretation naturally accounts for the observed Ising critical behavior. Furthermore, when the Zwanzig fluid is extended to more allowed particle orientations, we argue that in some cases the symmetry of a q-state Potts model with q>2 arises. This observation is used to interpret a number of previous results.

The particle spectrum of the three-state Potts field theory: a numerical study

Abstract: The three-state Potts field theory in two dimensions with thermal and magnetic perturbations provides the simplest model of confinement allowing for both mesons and baryons, as well as for an extended phase with deconfined quarks. We study numerically the evolution of the mass spectrum of this model over its whole parameter range, obtaining a pattern of confinement, particle decay and phase transitions which confirms recent predictions.

Restricted orientation "liquid crystal" in two dimensions: Isotropic-nematic transition or liquid-gas one(?)

Abstract: We present Monte Carlo simulation results of the two-dimensional Zwanzig fluid, which consists of hard line segments which may orient either horizontally or vertically. At a certain critical fugacity, we observe a phase transition with a two-dimensional Ising critical point. Above the transition point, the system is in an ordered state, with the majority of particles being either horizontally or vertically aligned. In contrast to previous work, we identify the transition as being of the liquid-gas type, as opposed to isotropic-to-nematic one. This interpretation naturally accounts for the observed Ising critical behavior. Furthermore, when the Zwanzig fluid is extended to more allowed particle orientations, we argue that in some cases the symmetry of a q-state Potts model with q>2 arises. This observation is used to interpret a number of previous results.

Some Remarks on a Generalization of the Superintegrable Chiral Potts Model

Abstract: The spontaneous magnetization of a two-dimensional lattice model can be expressed in terms of the partition function W of a system with fixed boundary spins and an extra weight dependent on the value of a particular central spin. For the superintegrable case of the chiral Potts model with cylindrical boundary conditions, W can be expressed in terms of reduced Hamiltonians H and a central spin operator S. We conjectured in a previous paper that W can be written as a determinant, similar to that of the Ising model. Here we generalize this conjecture to any Hamiltonians that satisfy a more general Onsager algebra, and give a conjecture for the elements of S.

Coexistence of Ordered and Disordered Phases in Potts Models in the Continuum

Abstract: This is the second of two papers on a continuum version of the Potts model, where particles are points in ℝ d , d≥2, with a spin which may take S≥3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ −1, γ>0. In this paper we prove phase transition, namely we prove that if the scaling parameter of the Kac potential is suitably small, given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist S+1 mutually distinct DLR measures.

Critical interfaces in the random-bond Potts model.

Abstract: We study geometrical properties of interfaces in the random-temperature $q$-states Potts model as an example of a conformal field theory weakly perturbed by quenched disorder. Using conformal perturbation theory in $q\ensuremath{-}2$ we compute the fractal dimension of Fortuin-Kasteleyn (FK) domain walls. We also compute it numerically both via the Wolff cluster algorithm for $q=3$ and via transfer-matrix evaluations. We also obtain numerical results for the fractal dimension of spin clusters interfaces for $q=3$. These are found numerically consistent with the duality ${\ensuremath{\kappa}}^{\mathrm{spin}}{\ensuremath{\kappa}}^{\mathrm{FK}}=16$ as expressed in putative SLE parameters.

From String Nets to Nonabelions

Abstract: We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an SO(3)3 × SO(3)3 doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.

The phase transition of the quantum Ising model is sharp

Abstract: An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called `random-parity' representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.

The $Z_2$ staggered vertex model and its applications

Abstract: New solvable vertex models can be easily obtained by staggering the spectral parameter in already known ones. This simple construction reveals some surprises: for appropriate values of the staggering, highly non-trivial continuum limits can be obtained. The simplest case of staggering with period two (the $Z_2$ case) for the six-vertex model was shown to be related, in one regime of the spectral parameter, to the critical antiferromagnetic Potts model on the square lattice, and has a non-compact continuum limit. Here, we study the other regime: in the very anisotropic limit, it can be viewed as a zig-zag spin chain with spin anisotropy, or as an anyonic chain with a generic (non-integer) number of species. From the Bethe-Ansatz solution, we obtain the central charge $c=2$, the conformal spectrum, and the continuum partition function, corresponding to one free boson and two Majorana fermions. Finally, we obtain a massive integrable deformation of the model on the lattice. Interestingly, its scattering theory is a massive version of the one for the flow between minimal models. The corresponding field theory is argued to be a complex version of the $C_2^{(2)}$ Toda theory.

Symmetry relations for multifractal spectra at random critical points

Abstract: Random critical points are generically characterized by multifractal properties. In the field of Anderson localization, Mirlin et al (2006 Phys. Rev. Lett. 97 046803) have proposed that the singularity spectrum f(α) of eigenfunctions satisfies the exact symmetry f(2d−α) = f(α)+d−α. In the present paper, we analyze the physical origin of this symmetry in relation to the Gallavotti–Cohen fluctuation relations of large deviation functions that are well known in the field of non-equilibrium dynamics: the multifractal spectrum of the disordered model corresponds to the large deviation function of the rescaling exponent γ = (α−d) along a renormalization trajectory in the effective time t = lnL. We conclude that the symmetry discovered for the specific example of Anderson transitions should actually be satisfied at many other random critical points after an appropriate translation. For many-body random phase transitions, where the critical properties are usually analyzed in terms of the multifractal spectrum H(a) and of the moment exponents X(N) of the two-point correlation function (Ludwig 1990 Nucl. Phys. B 330 639), the symmetry becomes H(2X(1)−a) = H(a)+a−X(1), or equivalently Δ(N) = Δ(1−N) for the anomalous parts . We present numerical tests favoring this symmetry for the 2D random Q-state Potts model with varying Q.

Mean Field Analysis of Low–Dimensional Systems

Abstract: For low–dimensional systems, (i.e. 2D and, to a certain extent, 1D) it is proved that mean–field theory can provide an asymptotic guideline to the phase structure of actual systems. In particular, for attractive pair interactions that are sufficiently “spead out” according to an exponential (Yukawa) potential it is shown that the energy, free energy and, in particular, the block magnetization (as defined on scales that are large compared with the lattice spacing but small compared to the range of the interaction) will only take on values near to those predicted by the associated mean–field theory. While this applies for systems in all dimensions, the significant applications are for d = 2 where it is shown: (a) If the mean–field theory has a discontinuous phase transition featuring the breaking of a discrete symmetry then this sort of transition will occur in the actual system. Prominent examples include the two–dimensional q = 3 state Potts model. (b) If the mean–field theory has a discontinuous transition accompanied by the breaking of a continuous symmetry, the thermodynamic discontinuity is preserved even if the symmetry breaking is forbidden in the actual system. E.g. the two–dimensional O(3) nematic liquid crystal. Further it is demonstrated that mean–field behavior in the vicinity of the magnetic transition for layered Ising and XY systems also occurs in actual layered systems (with spread–out interactions) even if genuine magnetic ordering is precluded.

Beyond the Boltzmann factor for corrections to scaling in ferromagnetic materials and critical fluids

Abstract: The Boltzmann factor comes from the linear change in entropy of an infinite heat bath during a local fluctuation; small systems have significant nonlinear terms. We present theoretical arguments, experimental data, and Monte-Carlo simulations indicating that nonlinear terms may also occur when a particle interacts directly with a finite number of neighboring particles, forming a local region that fluctuates independent of the infinite bath. A possible mechanism comes from the net force necessary to change the state of a particle while conserving local momentum. These finite-sized local regions yield nonlinear fluctuation constraints, beyond the Boltzmann factor. One such fluctuation constraint applied to simulations of the Ising model lowers the energy, makes the entropy extensive, and greatly improves agreement with the corrections to scaling measured in ferromagnetic materials and critical fluids.

A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees

Abstract: We give a criterion for the non-reconstructability of tree-indexed $q$-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix $M$. Non-reconstruction holds if $c(M)$ times the expected number of offspring on the Galton-Watson tree is smaller than 1. Here $c(M)$ is an explicit function, which is convex over the set of $M$'s with a given invariant distribution, that is defined in terms of a $(q-1)$-dimensional variational problem over symmetric entropies. This result is equivalent to proving the extremality of the free boundary condition Gibbs measure within the corresponding Gibbs-simplex. Our theorem holds for possibly non-reversible $M$ and its proof is based on a general recursion formula for expectations of a symmetrized relative entropy function, which invites their use as a Lyapunov function. In the case of the Potts model, the present theorem reproduces earlier results of the authors, with a simplified proof, in the case of the symmetric Ising model (where the argument becomes similar to the approach of Pemantle and Peres) the method produces the correct reconstruction threshold), in the case of the (strongly) asymmetric Ising model where the Kesten-Stigum bound is known to be not sharp the method provides improved numerical bounds.

Spanning Forests on Random Planar Lattices

Abstract: The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q→0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n→−1, the expansion parameter t counting the number of components of the forest.

Some exact results on the Potts model partition function in a magnetic field

Abstract: We consider the Potts model in a magnetic field on an arbitrary graph G. Using a formula by F Y Wu for the partition function Z of this model as a sum over spanning subgraphs of G, we prove some properties of Z concerning factorization, monotonicity and zeros. A generalization of the Tutte polynomial is presented that corresponds to this partition function. In this context, we formulate and discuss two weighted graph-coloring problems. We also give a general structural result for Z for cyclic strip graphs.